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Symbolic Powers of Monomial Ideals

Part of: General commutative ring theory Projective and enumerative geometry Arithmetic rings and other special rings

Published online by Cambridge University Press:  13 June 2016

Susan M. Cooper ,
Robert J. D. Embree ,
Huy Tài Hà  and
Andrew H. Hoefel
Susan M. Cooper
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA ([email protected])
Robert J. D. Embree
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada ([email protected]; [email protected])
Huy Tài Hà
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA ([email protected])
Andrew H. Hoefel
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada ([email protected]; [email protected])
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Abstract

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x 0, … , xn ] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

Keywords

symbolic powerssymbolic polyhedroninitial degreesmonomial ideals

MSC classification

Secondary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials 13A02: Graded rings 14N05: Projective techniques
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 39 - 55
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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